The rest of the paper is organized as follows. In Sec.  we describe the Lagrangian of the Goldberger-Wise Randall-Sundrum model and set notation for the background geometry. In Sec.  we describe the spin-0 and spin-2 mode expansions. Our analysis of the Kaluza-Klein expansions for this system follows the computations of [1]}, [2]}, [3]}, [4]}, generalized to de Donder gauge, and is presented in detail for completeness and clarity in Appendix . A review of Kaluza-Klein mode scattering and couplings, and description of the version of the sum rules of [5]} used here, as well as a description of the analytic proof of one combination of the sum rules involving the scalar sector is given in Sec. . Details of the analytic proof of the sum rule is given in Appendix REF . We also provide, in the totality of Appendix , a complete list of the sum-rule relations which must be satisfied for all \(2 \rightarrow 2\)  massive spin-2 scattering amplitudes to grow no faster than \({\cal O}(s)\)  – completing the analyses begun in [6]}, [7]}, [8]}, [9]}. A description and the analysis of the perturbative warped-stabilized model is given in Sec. . In particular, our numerical checks of the sum rules in this model are illustrated in Fig. REF  and Fig. REF  of Sec. REF . Our perturbative analysis requires a slight generalization of Rayleigh-Schrödinger perturbation theory to account for perturbations in the weight-function of the corresponding Sturm-Liouville problem, and this formalism is described in Appendix . Our conclusions are given in Sec. . Mathematica [10]} files giving the expressions for all the spin-2 and spin-0 perturbative wavefunctions can be found on GitHub.https://github.com/kirtimaan/Stabilized-Extra-Dimenion
